associated bundle construction
Let be a topological group, a (right) principal -bundle,
a topological space
and a
representation of as homeomorphisms of . Then the fiber bundle
associated to by , is a fiber bundle with fiber and group that is defined as follows:
- •
-
•
The projection is defined by
where denotes the –orbit of .
Theorem 1.
The above is well defined and defines a –bundle over with fiber
. Furthermore has the same transition functions as .
Sketch of proof.
To see that is well defined just notice that for and
, . To see that the fiber is notice that since
the principal action is simply transitive, given any orbit of the
–action on contains a unique representative of the
form for some . It is clear that an open cover that
trivializes trivializes as well.
To see that has the same
transition functions as notice that transition functions of act on the
left and thus commute with the principal –action on .
∎
Notice that if is a Lie group, a smooth principal bundle
and is a
smooth manifold and maps inside the diffeomorphism group of , the
above construction produces a smooth bundle. Also quite often has extra
structure
and maps into the homeomorphisms of that preserve that
structure. In that case the above construction produces a “bundle of such
structures.” For example when is a vector space and
, i.e. is a linear
representation of we get
a vector bundle
; if we get an
oriented vector bundle, etc.
Title | associated bundle construction |
---|---|
Canonical name | AssociatedBundleConstruction |
Date of creation | 2013-03-22 13:26:46 |
Last modified on | 2013-03-22 13:26:46 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 9 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 55R10 |
Defines | associated bundle |